Table Of ContentRELIABILITY AND
DECISION MAKING
RELIABILITY AND
DECISION MAKING
Edited by
RICHARD E. BARLOW
University of California,
Berkeley, California, USA
CARLO A. CLAROTTI
ENEA
(National Committee for New Technologies,
Energy and Environment, Rome, Italy)
FABIO SPIZZICHINO
University of Rome-La Sapienza,
Rome, Italy
1m
SPRINGER-SCIENCE+BUSINESS MEDIA, B.Y.
First edition 1993
© Springer Science+Business Media Dordrecht 1993
Originally published by Chapman & Hall in 1993
Softcover reprint of the hardcover 1st edition 1993
Typeset in Malta by Interprint Limited
ISBN 978-0-412-53480-5 ISBN 978-1-4899-4459-7 (eBook)
DOI 10.1007/978-1-4899-4459-7
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Preface
This Volume gathers together papers given at the conference held in
Siena, Italy, from October 15 through October 26, 1990. The aim of the
Volume is more ambitious than just to provide a published account of
the proceedings of the conference. The aim is to supply the reliability
community with a coherent Bayesian predictive view on reliability theory
and practice.
We may ask:
what is the predictive Bayesian approach to reliability?
what makes the predictive Bayesian approach to reliability a
preferred, if not a compulsory, choice?
We shall try to make a long story short. The editors and most of the
contributors to this Volume have recognised that the ultimate purpose of
reliability analysis is to take decisions relative to engineering systems in
the face of uncertainty. In short:
any system of concern is prone to failure no matter which one of
the conceivable design-solutions is implemented;
system failure (success) will cause a loss (reward) to be incurred
(cashed).
The following decision problem is imposed upon the decision maker:
which design solution should be implemented? Note that 'keeping the
status quo', i.e. not building the system, is not a 'conservative' solution.
Any decision-maker (a business-man, a producer or other) who system
atically misses out on the chance of earning money, just because there is
a remote possibility of losing money, will eventually go bankrupt. The
decision maker needs a measure of his uncertainty to assess whether or
not the possibility of failure is 'remote enough' with respect to the
possible loss and the possible reward. He acts then in a betting frame
work with the restriction that he will not consider a bet which promises
certain loss. This restriction is the principle which forces us to assess
uncertainty by probability and to use the calculus of probability in
making decisions. No other measure of uncertainty guarantees the
decision maker that his bet will never result in certain loss.
v
vi Preface
Adopting the predictive Bayesian approach to reliability means that we
must coherently apply the laws of subjective probabilities to reliability
decision problems. This approach requires radical changes in current
probabilistic/statistical techniques used in reliability practice. In particular:
I. Only observable quantities are the 'random variables par excellence':
parameters of probability distributions are just a computational aid for
deriving the conditional probability of future observations given the
values of the past ones.
II. In any reliability decision problem, the related utility function must be
a function of the random variables (e.g. the failure time of the system) whose
actual values will cause the loss to be incurred or the reward to be cashed.
The utility function cannot depend upon abstract variables (such as the
failure rate) whose value is still unknown when the system has failed or
accomplished its mission. In view of the Maximum Expected Utility (MEU)
principle, taking reliability decisions via (point or interval) estimation of
probability-distribution-parameters does not make then any sense.
III. Stochastic dependence must be understood as synonymous with
the possibility of learning and not necessarily as synonymous with
physical dependence.
IV. Any stochastic model must be expressed in terms of the joint
probability distribution for the random variables of interest. Exchangea
bility may playa central role in this analysis.
V. The probabilistic concept of aging must be rethought.
VI. In deriving the joint distribution of any set of life-times, one must
use an approach which accounts for the dynamical character of the
information.
VII. The methods of optimal control theory are often needed. The use
of such methods is needed whenever the statistical decision problem of
concern is sequential in continuous time. Reliability decision problems
are usually continuous time decision problems and, in the most general
case, they are sequential.
Adopting the predictive Bayesian approach to reliability is (in our
view) not only compelling, but also convenient. It is compelling in that it
is the only way to avoid 'Dutch Books'. It is convenient because it
simplifies and unifies several major reliability problems. In a Bayesian
predictive framework:
there is no substantial difference between a system reliability
problem and a problem of statistical analysis of failure and survival
data observed in the past;
Preface vii
all the well known bounds on system reliability can be exploited
also when probability distribution parameters are unknown;
burn-in and life-testing can be regarded as particular design
solutions of a more general decision problem;
preferability for different decisions can be stated without condition
ing on the value of some non-observable quantity;
the terms 'uncertainty about the model' and 'uncertainty about the
values of probabilities' can be rephrased in a rational way. This
leads to the recognition that there is no difference between the 'two
types of uncertainty'.
The volume has been divided into four sections:
1. Fundamentals of Statistical Decision Theory;
2. Sequential Problems in Reliability: a Dynamical Approach;
3. Bayesian Models in Reliability and Quality Control;
4. Engineering Reliability.
This has been done to permit selective reading. Those who are only
interested in becoming acquainted with the motivation for the Bayesian
predictive approach to reliability, and with its implementation in the case in
which there is no forthcoming information can just look at Sections 1 and 4.
In Section 1, the following issues are addressed:
frequentists claim that they base their decisions only on the solid
objective grounds of data; this claim is false since in orthodox
statistics the likelihood principle is violated (see the papers by
Clarotti and by Piccinato);
avoidance of Dutch Books is attained if subjective probability is
used for making decisions. This is shown in the paper by Bruce Hill
where in addition he discusses: (1) how to use theorems derived for
unending sequences of random variables when dealing with finite
sets of random variables; and (2) the role of sensitivity analysis in
the Bayesian approach;
the influence-diagrams tool is the 'user-friendly' (see the article by
Barlow and Pereira) version of the predictive Bayesian approach to
reliability.
Section 2 is rather technical but it is at the core of the Bayesian
predictive approach:
as stated any reliability problem is a decision problem in the face of
uncertainty;
viii Preface
uncertainty can be mitigated by new evidence; the latter has in
general a dynamical character;
often one has some control over the development of new informa
tion;
the first decision to be considered is: what actions will result in
new information which is optimal in the sense of expected cost?
Optimal stopping of Markov processes is used by Spizzichino to answer
this question when in particular one has to decide the sampling plan of a
batch of new similar units.
in solving this problem, the central role of the concept of Multi
variate Conditional Failure Rate introduced by Shaked and
Shanthikumar becomes evident;
more generally the need of answering the question above, and
the dynamical character of statistical evidence, requires that
theories of stochastic control and of point processes now play
an important role in the reliability theory. The reader is given
an insight into these disciplines and into their reliability applica
tions by the papers by Arjas, Gerardi and Koch, and Runggaldier.
In Section 3 everyone (we hope) can find at least one Bayesian model
specific to his/her own field of interest:
Barlow and Mendel discuss a new probabilistic notion of aging
based on the Bayesian approach;
Polson, Singpurwalla and Verdinelli discuss designs for acceler
ated life tests; they describe how an inferential problem can be
tackled as a decision problem (remember anyway that a reliabi
lity decision problem cannot be reconducted to an inferential
problem);
Barlow and Irony discuss a Bayesian approach to quality control;
Singpurwalla discusses Taguchi methods from a Bayesian decision
theory point of view;
Muliere and Scarsini illustrate the change point problem and its
role in quality control.
In Section 4 the theory of coherent structures is considered from
different points of view. This makes it apparent that:
in a frequentist frame system optimisation can be achieved when
the related probability-distribution-parameters are completely
specified (Boland discusses majority voting in this context);
Preface ix
the approach based on the Bayesian estimation of parameters not
only violates the MEV principle but also requires a great deal of
mathematical skill to produce results valid under stringent hy
potheses (e.g. series system formed from non-identical components,
as shown in the paper by Gertzbach and Kordonski);
the predictive Bayesian approach is the 'natural' companion of the
fault-tree technique (Clarotti).
In Section 4 furthermore
Apostolakis and Wu review the debate in the safety engineering
community concerning the interpretation of probability;
the intriguing matter of prior knowledge in engineering applica
tions is discussed by Natvig.
We hope this volume succeeds in providing the information necessary
to use the Bayesian predictive approach in reliability.
Richard E. Barlow, Carlo A. Clarotti, Fabio Spizzichino
Acknowledgements
The editors efforts to circulate Bayesian predictive ideas about reliability
theory and practice would have been useless had the conference 'Reliabil
ity and Decision Making' not been held. The Conference was directed by
C. A. Clarotti, G. Koch and F. Spizzichino and was organised under the
auspices and the financial support of University of Siena and of Comitato
Nazionale per Ie Scienze Matematiche of C.N.R. (National Board for
Science in Italy). Particular thanks are to be expressed to Prof. Luigi
Berlinguer (Rector of the University of Siena), to Prof. Carlo Ciliberto
(President of Comitato Nazionale per Ie Scienze Matematiche of C.N.R.)
and to Prof. Carlo Cercignani (President of the Subcommittee for
Applied Mathematics of C.N.R.). Their sensibility for initiatives in the
field of applied mathematics made it possible to mobilise resources for the
Conference. The Editors are also much in-debted to their friend Giorgio
Koch for acting Co-director of the Conference and to Prof. Vincenzo
Millucci (Dept. of Mathematics, University of Siena) who strongly
pleaded the cause of the Conference at University of Siena.
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